Understanding Population Dynamics Using Partial Differential Equations

Serena Wang, Gargi Mishra, Caledonia Wilson, Michelle Serrano

Pierre-Francois Verhulst came up with the logistic model

Thomas Malthus Pierre-Francois Verhulst

Logistic Model

y’ = ky(A-y)

k: the population growth constant

A: the carrying capacity of the environment

General solution:

What are Partial Differential Equations?

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

http://www.its.caltech.edu/~sparikh/ma142.html

Time Evolution of a Spatial Profile Curve

Our Model:

● Modelling population over a thin strip of habitat ● Want to model population as a function of location and time along

this strip● Let u(t,x) denote the size of a population located at a location x for a

particular time t

● Then our initial conditions take the form u(0,x)=u0(x)

The Initial Profile Curve, u0(x)

Changing the Logistic Equation

In our new model, the carrying capacity, A, now depends on location, x

Location --> Location -->

Pop

ulat

ion

--->

Pop

ulat

ion

--->

Why use partial differential equations?

● Allows us to conceptualize how the initial profile curve changes with time: “the time evolution of the profile curve”

● Breaking down problem like this allows us to model migration and dispersion: we focused on migration

How u(t,x) changes with time when we let x be constant

How the curve u0(x) is modified over time

Mathematica Plot: Time Evolution of the Profile Curve

Stability of Equilibria

Migrating Populations

c: “speed” of migrationx: Locationt: timeu(t,x): function of time and location

Initial Condition:

PREVIOUS EXAMPLE

NEW EXAMPLE

Migrating Populations: A General Solution for the PDE

Integrating both sides of this equation with respect to t:

Gives:

Derivation of the Equation

Take partial derivative of:

To Get:

x

ut=0 t=1

-c*t

Mathematica Plot: A randomly migrating population

Case Study: Seasonal Migration

- In our model, we use migration of the gray whales to simulate the behavior of the function.

- The migration of gray whales is seasonal, ranging from the Southern Baja peninsula near the Tropic of Cancer to the Chukchi Sea north of the Arctic Circle.

- To model the periodic nature of the whales’ migration, we can translate the initial graph using a periodic function to obtain

Periodic Migration

Now, u(t,x) satisfies the periodic migration equation

Where u0(x) acts as the initial profile curve. The parameter ‘c’ still

governs the speed of the migration dependent on the time of

year.

Mathematica Plot: Movement of Curve

References

Kerckhove, Michael. "From Population Dynamics to Partial Differential Equations." The Mathematica Journal 14 (2012). doi:10.3888/tmj.14-9.

Thank You! Questions?